Theorem finlocfin 23528

Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
finlocfin.1	⊢ 𝑋 = ∪ 𝐽
finlocfin.2	⊢ 𝑌 = ∪ 𝐴

Assertion

Ref	Expression
finlocfin	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Proof of Theorem finlocfin

Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐽 ∈ Top)
2		simp3 1139	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3		simpl1 1192	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Top)
4		finlocfin.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 22912	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6	3, 5	syl 17	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ 𝐽)
7		simpr 484	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8		simpl2 1193	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ Fin)
9		ssrab2 4080	. . . . 5 ⊢ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴
10		ssfi 9213	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
11	8, 9, 10	sylancl 586	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)
12		eleq2 2830	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑥 ∈ 𝑛 ↔ 𝑥 ∈ 𝑋))
13		ineq2 4214	. . . . . . . . 9 ⊢ (𝑛 = 𝑋 → (𝑠 ∩ 𝑛) = (𝑠 ∩ 𝑋))
14	13	neeq1d 3000	. . . . . . . 8 ⊢ (𝑛 = 𝑋 → ((𝑠 ∩ 𝑛) ≠ ∅ ↔ (𝑠 ∩ 𝑋) ≠ ∅))
15	14	rabbidv 3444	. . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅})
16	15	eleq1d 2826	. . . . . 6 ⊢ (𝑛 = 𝑋 → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin))
17	12, 16	anbi12d 632	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)))
18	17	rspcev 3622	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
19	6, 7, 11, 18	syl12anc 837	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
20	19	ralrimiva 3146	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
21		finlocfin.2	. . 3 ⊢ 𝑌 = ∪ 𝐴
22	4, 21	islocfin 23525	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
23	1, 2, 20, 22	syl3anbrc 1344	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ‘cfv 6561  Fincfn 8985  Topctop 22899  LocFinclocfin 23512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989  df-top 22900  df-locfin 23515

This theorem is referenced by:  locfincmp  23534  cmppcmp  33857